Pinning via core interaction occurs due to minimization of the core energy of the vortices when they encounter pinning centers [17, 18]. The vortex core energy saved when a vortex encounters a pinning center depends on the geometry of the pinning center and degree of suppression of the order parameter. Pinning force of a pinning center depends on the gradient of the energy variation within the pinning center. If the pinning center is much larger than the vortex core (i.e., £), it will be a weak pinning center. This is because the energy of the vortex will change very gradually as it passes the pinning center. On the other hand, if it is much smaller than it cannot contribute substantially to lowering the energy of the system and the energy gradient is again small, even though the variation occurs on a much smaller scale. Therefore, the most effective pinning centers are of the size of
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